x [ n ] = eJnnn= cos R o n +j sin R o n in .NET framework

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x [ n ] = eJnnn= cos R o n +j sin R o n
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(1.53)
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Thus x [ n ] is a complex sequence whose real part is cos R o n and imaginary part is sin R o n .
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In order for ejn@ to be periodic with period N ( > O), R o must satisfy the following condition (Prob. 1.1 1):
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m = positive integer
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Thus the sequence eJnonis not periodic for any value of R,. It is periodic only if R , / ~ I T is a rational number. Note that this property is quite different from the property that the continuous-time signal eJwo' is periodic for any value of o . Thus, if R, satisfies the , periodicity condition in Eq. (1.54), ! f 0, and N and m have no factors in common, then & the fundamental period of the sequence x[n] in Eq. (1.52) is No given by
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Another very important distinction between the discrete-time and continuous-time complex exponentials is that the signals el"o' are all distinct for distinct values of w , but that this is not the case for the signals ejRon.
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SIGNALS AND SYSTEMS
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[CHAP. 1
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Fig. 1-12
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Real exponential sequences. (a) a > 1; ( b ) 1 > a > 0; (c) 0 > a > - 1; ( d l a <
- 1.
CHAP. 11
SIGNALS AND SYSTEMS
Consider the complex exponential sequence with frequency (R, integer:
ej(ilo+2rrk)n
+ 2 7 k ) , where k
is an
= e j R o nej2rrkn
- ei n o n -
(1.56)
since e j 2 " k n = 1. From Eq. (1.56) we see that the complex exponential sequence at frequency R, is the same as that at frequencies (R, f 27), (R, f 4.~1, and so on. Therefore, in dealing with discrete-time exponentials, we need only consider an interval of length 2 7 in which to choose R,. Usually, we will use the interval 0 I R, < 2 7 or the interval - 7 sr R, < 7 .
General Complex Exponential Sequences:
The most general complex exponential sequence is often defined as x [ n ] = Can
(1.57)
where C and a are in general complex numbers. Note that Eq. (1.52) is the special case of Eq. (1.57) with C = 1 and a = eJRO.
.. .
Ylll
'111'
= cos(rrn/6);
Fig. 1-13 Sinusoidal sequences. (a) x[n]
( b )x[n] = cos(n/2).
SIGNALS AND SYSTEMS
[CHAP. 1
Real Exponential Sequences:
If C and a in Eq. (1.57) are both real, then x[n] is a real exponential sequence. Four distinct cases can be identified: a > 1 , 0 < a < 1, - 1 < a < 0, and a < - 1. These four real exponential sequences are shown in Fig. 1-12. Note that if a = 1, x[n] is a constant sequence, whereas if a = - 1, x[n] alternates in value between + C and -C. D. Sinusoidal Sequences: A sinusoidal sequence can be expressed as
If n is dimensionless, then both R, and 0 have units of radians. Two examples of sinusoidal sequences are shown in Fig. 1-13. As before, the sinusoidal sequence in Eq. (1.58) can be expressed as
As we observed in the case of the complex exponential sequence in Eq. (1.52), the same observations [Eqs. (1.54) and (1.5611 also hold for sinusoidal sequences. For instance, the sequence in Fig. 1-13(a) is periodic with fundamental period 12, but the sequence in Fig. l-13( b ) is not periodic.
SYSTEMS AND CLASSIFICATION OF SYSTEMS System Representation:
A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal. Let x and y be the input and output signals, respectively, of a system. Then the system is viewed as a transformation (or mapping) of x into y. This transformation is represented by the mathematical notation
where T is the operator representing some well-defined rule by which x is transformed into y. Relationship (1.60) is depicted as shown in Fig. 1-14(a). Multiple input and/or output signals are possible as shown in Fig. 1-14(b). We will restrict our attention for the most part in this text to the single-input, single-output case.
System
Sy stem
Fig. 1-14 System with single or multiple input and output signals.
CHAP. 11
SIGNALS AND SYSTEMS
B. Continuous;Time and Discrete-Time Systems:
If the input and output signals x and p are continuous-time signals, then the system is called a continuous-time system [Fig. I - 15(a)].If the input and output signals are discrete-time signals or sequences, then the system is called a discrete-time s .stem [Fig. I - 15(h)J.
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